On the Distribution of Three-Term Arithmetic Progressions in Sparse Subsets of Fpn

We give a short proof for the following result on the distribution of three-term arithmetic progressions in sparse subsets of Fp : for every α > 0 there exists a constant C = C(α) such that the following holds for all r ≥ Cp and for almost all sets R of size r of Fp . Let A be any subset of R of size at least αr, then A contains a non-trivial three-term arithmetic progression. This is an analog of a hard theorem by Kohayakawa, Luczak, and Rödl. The proof uses a version of Green’s regularity lemma for subsets of a typical random set, which is of interest of its own.